Problem: Solve for $x$ : $ 8|x + 9| + 1 = -3|x + 9| + 4 $
Solution: Add $ {3|x + 9|} $ to both sides: $ \begin{eqnarray} 8|x + 9| + 1 &=& -3|x + 9| + 4 \\ \\ { + 3|x + 9|} && { + 3|x + 9|} \\ \\ 11|x + 9| + 1 &=& 4 \end{eqnarray} $ Subtract ${1}$ from both sides: $ \begin{eqnarray} 11|x + 9| + 1 &=& 4 \\ \\ { - 1} &=& { - 1} \\ \\ 11|x + 9| &=& 3 \end{eqnarray} $ Divide both sides by ${11}$ $ \dfrac{11|x + 9|} {{11}} = \dfrac{3} {{11}} $ Simplify: $ |x + 9| = \dfrac{3}{11}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 9 = -\dfrac{3}{11} $ or $ x + 9 = \dfrac{3}{11} $ Solve for the solution where $x + 9$ is negative: $ x + 9 = -\dfrac{3}{11} $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& -\dfrac{3}{11} \\ \\ {- 9} && {- 9} \\ \\ x &=& -\dfrac{3}{11} - 9 \end{eqnarray} $ Change the ${ - 9}$ to an equivalent fraction with a denominator of $11$ $ x = - \dfrac{3}{11} {- \dfrac{99}{11}} $ $ x = -\dfrac{102}{11} $ Then calculate the solution where $x + 9$ is positive: $ x + 9 = \dfrac{3}{11} $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& \dfrac{3}{11} \\ \\ {- 9} && {- 9} \\ \\ x &=& \dfrac{3}{11} - 9 \end{eqnarray} $ Change the ${ - 9}$ to an equivalent fraction with a denominator of $11$ $ x = \dfrac{3}{11} {- \dfrac{99}{11}} $ $ x = -\dfrac{96}{11} $ Thus, the correct answer is $x = -\dfrac{102}{11} $ or $x = -\dfrac{96}{11} $.